[Proof of Proposition REF ]
The proof follows verbatim that of Theorem 8.5 in [1]}, with the exception of the fact that exact \(p\) -values are replaced by approximate \(p\) -values, including the rate of approximation. We included the full proof for the convenience of the reader. By the definition of the FDR,
\(\mathrm {FDR}(\hat{\tau })=\Big [\frac{ \sum _{(t, k) \in _{0}} [|\tilde{W}_{t,k}| > \hat{\tau }] [R_{\hat{\tau }} > 0]}{R_{\hat{\tau }}}\Big ]=\sum _{(t, k) \in _{0}}\Big [\frac{[|\tilde{W}_{t,k}| > \hat{\tau }] [R_{\hat{\tau }} > 0]}{R_{\hat{\tau }}}\Big ].\) 
